A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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upper bound for Ito integral of deterministic integrand

It is well known that Ito integrals with respect to a Brownian motion cannot be defined pathwise because the Brownian motion has infinite 1st order variation. These integrals are defined as limits of ...
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17 views

Law and Brownian Bridge

Let $Z_{t}= W_{t}-tW_{1}$ and $Y_{1}=\sup_{0\leq t\leq 1}Z_{t}$, $(W_t, t \geq 0)$ standard Brownian motion Find the law of $Y_{1}$ I know that $\textbf{P}(\sup_{0\leq t\leq 1}W_{t}\geq x , ...
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10 views

Does an integrable IID continuous time stochastic process exist?

Let $t\in[0,T)$ where $0 < T \leq \infty$, and assume a stochastic process exists $Z_t$. The question is: does there exist an IID stochastic process for $Z_t$ such that $Z_t \perp Z_{\tau}$ for ...
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1answer
15 views

Ito integrals and the Euler scheme

I was wondering how to find the solution of the following stochastic integral: $$dY_{t}=a(W_{t},Y_{t})dW_{t}+b(W_{t},Y_{t})dZ_{t}$$ or in integral notation ...
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13 views

Ultrametric space of stochastic filtration

Let $\Omega$ be an arbitrary set and $(\mathscr F_t)_{t\in \Bbb R_+}$ be a non-decresing sequence of $\sigma$-algebras on $\Omega$ such that any subset of $\Omega$ is contained is some of them, that ...
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11 views

Mutual information staying constant under composition of channels

Consider the following scenario: one has 2 communication channels $C_1$ and $C_2$. Let $p_0(x)$ be some arbitrary but fixed input probability distribution. The mutual information between the input ...
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55 views

Upper bounding a Poisson Process with indicators of exponentials

Define $E_1,E_2,\ldots, E_i,\ldots E_n$ as i.i.d. exponentials with parameter $\lambda$. These define processes on some interval $[0,\delta]$ (think of $\delta$ as very small, it will come into play ...
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1answer
45 views

Exponential of Squared Brownian Motion

Long time lurker, first time posting! Have a problem, that looks familiar but I can't put my finger on it. Need to calculate $\mathbb{E} [\exp(aW_T^2)|F_t]$ where $W_t$ is an $F_t$ adapted standard ...
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1answer
24 views

Calculating probability of a time-series probability crossing a threshold

(Please feel free to suggest a better title -- I'm still not sure what to call this in the first place.) I'm having trouble getting my head wrapped around a time-series stochastics problem I've run ...
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1answer
16 views

Find the probability $P[ x(t) \le 1]$ where $x(t)$ is a filtered Poisson process (rect pulses)

I can't understand the following question: "The random process x(t) is defined as $$x(t) = \sum_{n=- \infty}^{+\infty} rect(\frac{t-\tau_{n}}{T}) \quad ,\quad t \ \epsilon \ (R)$$ where {$\tau_{n}$} ...
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25 views

Can the transition probabilities of an inhomogeneous Markov chain be written as an exponential?

If $Z_t$ is a homogeneous continuous-time Markov chain with finite state space $E=\{1,\ldots,p\}$, transition matrices $(P(t))$ and intensity matrix $Q$, it holds that $$ P(t) = \exp(tQ), $$ see for ...
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71 views

Conditional expectation of the sum of two random variables

I've got some difficulties in calculating the conditional expectation of the sum of two RV. I am not sure if I correctly formalized the scenario I am looking at. So I am trying to describe it first: ...
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10 views

Determining Moving-Average Representation of AR(2) Process

Consider a stationary $AR(2)$ process given by $$X_{t} - X_{t-1} + 0.25X_{t-2} = 5 + a_{t}$$ where $a_{t} \sim WN(0,1)$ (white noise). I am interested in obtaining the causal representation of ...
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23 views

Matlab code for higher order scheme

Can somebody help me how to generate the code for the increment $\Delta$Z in the document I have attached? I know how to generate the rest of the increments but struggling in how to generate ...
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34 views

Maps that preserve Brownian motion law

I am looking for a list of maps that take Brownian motion to Brownian motion: Here are some: Any rigid transformation ...
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1answer
41 views

Measurability question for càglàd process

I have encounter the following question, which is probably naive for probabilists, but let me still ask it: Let $\{X_t(\omega): t \in \mathbb{R}\}$ be a real valued càglàd process (that is ...
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2answers
33 views

Resolvent operators and inverses proof

I am trying to prove for myself that $A(R_{\alpha}g)=\alpha R_{\alpha}g-g$ which is proving problematic. The definition of $A$, the generator, is $\displaystyle Af(x)= \lim_{t \rightarrow 0} ...
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25 views

Question on generators in the proof of Kolmogorov's Backward Equation

Here is a part of the proof of the Kolmogorov's Backward Equation. I cannot see why $Y_t$ has been picked as it has. In particular, I cannot see why you would want to subtract t in the first bit of ...
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17 views

Milestein Scheme

Im struggling in the following schemes. I cant understand how the first scheme is equivalent to the second one. Can somebody help me? Thanks in advance. Moreover there is a typo error in the ...
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1answer
28 views

Finding the probability of ever visiting a transient state for a zero-seeking device for a Markov Chain?

A zero-seeking device operates as follows: if it is in state $j$ at time $n$, then at time $n+1$, its position is $0$ with probability $\frac{1}{j}$ or $k$ with probability $\frac{2k}{j^2}$, where $k$ ...
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1answer
16 views

Producing a transient & recurrent example for a birth-death chain with a more complex relationship between the $p_x$ and $q_x$.

Suppose we have a birth-death chain with a state space $$ S = \{0,1,2,\ldots\} $$ and transition probailities: $$p(x,y)=\begin{cases}q_x, &\text{if } y = x-1, &\text{i.e. death}\\ ...
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23 views

A question about Markov

There is a continuous-time markov chain,and we know the probability transition matrix P.The time between 2 states can be formulated as a exponential distribution whose u is related to the 2 states.Now ...
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7 views

Stationarity of Hawkes processes with (partially) negative kernels

Consider a point process $N$. For the linear Hawkes process, the conditional intensity is given by $\lambda(t) = \nu + \int h(t-s) N(ds)$, with constant $\nu > 0$ and kernel $h(s)$. In almost every ...
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1answer
19 views

Likelihood function of a Poisson process

Fix a window of time $[0,T]$ and say that we get $n$ arrival times in the window from a homogeneous Poisson process. The maximum likelihood estimate (MLE) is just $n/T$ I believe. But what is the ...
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26 views

Transition kernel that is not Markov

Let $(X,\mathcal{F})$ and $(Y,\mathcal{G})$ be two measurable space. A transition kernel $K$ is a function $K : X \times \mathcal{G} \to \overline{\mathbb{R}}_+$ suche that $K(\cdot,B)$ is measurable ...
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1answer
34 views

CDF for non-homogeneous Poisson process [duplicate]

I am trying to understand the inverse transform method for simulating random processes and have managed to completely confuse myself. Consider a Poisson process whose conditional intensity is ...
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14 views

The definition of spectral density of stationary process through Fourier transform

I recently took up studying elementary stochastic control theory and I have trouble comprehending why exactly is the (cross) spectral density defined in many texts as ...
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1answer
50 views

Prove Number of Arrivals $N(s)$ up to time $s$ follows $\mathrm{Poisson}(\lambda s)$ Distribution

This comes from my self-study of Durrett's "Essentials of Stochastic Processes" book, page 97. Definition Let $\tau_1,\tau_2,\ldots$ be independent $\mathrm{exponential}(\lambda)$ random variables. ...
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7 views

M/D/infinity queue - Steady state probability distribution of number of busy servers

Is there any work already available which gives steady state pdf of number of busy servers for M/D/infinity queue? If not, could any one help me in deriving the same. Thanks Seth
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62 views

Response of stationary distribution to perturbation of a stochastic matrix

Suppose I have a (left) stochastic matrix $P$, i.e. a non-negative matrix with column sums equal to 1. Its maximum eigenvalue will be equal to 1, and the corresponding eigenvector $\mathbf q$, if it's ...
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1answer
107 views

Deducing an optimal gambling strategy (using martingales).

Apologies in advance for the length, I tried being precise. Suppose a game where in each turn you can gamble a certain amount of money on the result of a fair coin toss. If the coin comes out tails ...
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40 views

More on the Existence and Uniqueness of the solutions of an SDE Proof

An extract from the proof of the existence and uniqueness of the solution of a SDE from Oksendal. I cannot see how holders inequality and the ito isometry are applied.
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22 views

Part of Proof of the Uniqueness of the Solution of SDE's

This is an extract from Oksendal's SDE of the proof of the uniqueness of the solution of a SDE. I cannot see how the $P[|X_t-\hat{X_t}|=0 \ \ \ \text{for all t} \in \mathbb{Q} \cap [0,T]]=1$ is ...
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1answer
23 views

Product of independent continuous local martingales is local martingale

Revuz-Yor's book mentioned if $M$ and $N$ are independent continuous local martingales, then $MN$ is still local martingale. But I don't know how to prove it. Any help, thanks!
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22 views

How to compute cumulative intensity process integral?

I am faced with a basic question about counting process and its intensity process used in survival analysis. It is actually the textbook example from Aalen's Survival and Event history analysis book. ...
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12 views

Error of a Serial Processs

Give random variable X and two processes A, B . Assume that $ Y_{1}, Y_{2}$ are estimated versions of X by using processes A, B respectively, with probability: $P\left \{ \left | X-Y_{1} \right ...
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1answer
18 views

Finding the unique Martingale Convergence Representation of a given r.v.

According to the martingale representation there exists a unique $g(t,\omega) \in \mathcal{V}(0,T)$ such that $M_t = E[M_0]+\int^{t}_{0} g(s,\omega) dB(s); \ \ \ t \in [0,T]$ Find g in the case ...
2
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1answer
29 views

derivation law from the call option formula

i am reading a article about the option pricing. and i got stuck with some typical statement. $C(K)=\int (x-K)^+\mu(dx)$ is given. here, $\mu$ is implied law of asset price and C(K) is the price ...
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19 views

If $C$ is finite, closed, irreducible, we know that every such state $C$ is recurrent. What is a counter example of if $C$ isn't closed? [on hold]

If $C$ is finite, closed, irreducible, we know that every such state $C$ is recurrent. What is a counter example of if $C$ isn't closed? I am trying to think of one but to no avail. Thank you! Here, ...
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1answer
31 views

martingale difference

I am trying to solve the following question. {$ξ_k$} is $F_n$-martingale difference (i.e. for every $n$, $E[ξ_n|F_{n-1}]=0 $ a.s. ) Also, for every $n$ , $E[ξ_n^2]<\infty$ Show that ...
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1answer
140 views

Stochastic inequality, true?

Consider two stochastic processes $X$ and $Y$ satisfying the following SDEs (with the same drift!): $$X_t = x + \int_0^t b(X_s)ds + B_t$$ $$Y_t = y + \int_0^t b(Y_s)ds + B_t.$$ If $0<x<y$, is ...
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1answer
33 views

More preliminaries of the Martingale Convergence Theorem

Really struggling with this lemma. Not sure about the general structure of the proof. Why have we chosen g to be orthogonal to all functions of the form 4.3.1? Why should $G(\lambda)=0$, does it ...
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22 views

Preliminaries of the Martingale Representation Theorem

I cannot understand why we are taking a dense subset of $[0,T]$. Furthermore, I cannot see a result that would allow each such $g_n(B_{t_1},\ldots,B_{t_n})$ to be approximated in ...
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19 views

Definition of Time Series

Having not done any stats for a few years, I seek clarification regarding the definition of time series given in my textbook. I apologize for the length, but I would be glad to just resolve my main ...
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1answer
45 views

Ito formula applied to $\frac{1}{t}\int_0^t W_s ds $

I got this expression and I have to calculate its differential by the Ito formula, $W_t$ denotes the Brownian motion: $$\frac{1}{t}\int_0^t W_s ds $$ I calculate the derivative of ...
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1answer
30 views

Evaluating Stratonovich integral from definition

I am struggling to evaluate the integral $\displaystyle \int^{T}_{0} B_t \circ dB_t $ from definition. So far I have that $\begin{align} \displaystyle \sum ...
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1answer
35 views

Measurability and knowledge

there seems to be a subtle relationship between knowledge and measurability. If I have a stochastic process $(X_n)_n$, then for example a stopping time ( other examples would be martingales, ...
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1answer
25 views

Q-matrix vs. P-matrix description of a Markov chain

Consider a continuous time Markov chain $(X_t)_{t \geq 0}$ on some state space $S$ with transition matrix (P-matrix) $p_t(x,y)$, the probability density of jumping from $x$ to $y$ in time $t$. The ...
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43 views

recurrent events-Probability of even number of successes

Let E be the event of an even number of successes. $u_n$:Probability of E occurring at the nth trial not necessarily for the first time $f_n$:Probability of E occurring at the nth trial for the first ...
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2answers
23 views

Probability of returning to a given state after n transitions-Markov chains

Let us denote $f_j^{(n)}$ denote the probability of the first return to state $j $after n transitions. Let $p_{jj}^{(n)}$ be the probability of returning to the state $j$ after $n$ transitions when ...